PREPARATION AND TRIAL OF THE
MATERIALS

The setting of the study is a rural school in Ponce
which is the second largest city of the island of Puerto Rico. Ponce is on the southern
coast. The school, which is situated on the lower part of a very hilly area, is
part of a system of two schools: a primary level (K-3) and a middle school
(4-6) level. The school population consists of 514 students and 80 teachers
with two principals.

The students who initially began in the study were
103 third grade students only two of whom did not take the posttest. The age
range was from 7 to 9 year old and there were 57 females and 46 males.

In
the community where the study took place ninety-nine percent of the families
receive government financial and medical aid.

Four out of eight third grade classes were selected
based on results of an achievement examination that was provided to the
investigator by the school district supervisor. The investigator administered
the test to the eight third grades and the selection of the sample was made
using the highest and the lowest mean scores on the achievement examination.
Two high and two low achiever groups were selected that best accommodated to
the school's program. The treatment (t1 and t2) and control groups (c1 and c2) each had a high (t2 and c2) and a low (t1 and c1) achiever group. Two
teachers participated in the study. One of which had three of the four groups (two
treatment and one control group) and the second teacher taught one control
group (see Table 1) .

The subjects for the clinical interviews from each
of the four groups were selected at random from two envelopes for each group. Each
envelope contained the names 5 students from the group, that attained scores
close to the mean and 5

who
attained scores of above one standard deviation from the mean. Two students
were selected from each of the two

envelopes
for each of the four groups for a total of 4 students per group. The total
number of students interviewed

The pre and posttest was designed using the criteria
of mental computation skills for addition and subtraction most commonly cited
in the literature. The identical test was used both as a pre-treatment and a
post-treatment measure. The examination consisted of three parts: one oral and
visual (using transparencies) consisting of nine short word problems,the second part was completely oral consisting of 25 exercises and a third part which was oral andvisual consisting 23 exercises similar
to those of the second part. As can be seen in the criteria table in Appendix
A, each item of the second part was matched to one of the third part to compare
the two modes of presentation. A first pilot examination was given in June 1995
at a private school in Ponce, Puerto Rico. The group consisted of 18 second
grade students. The second grade was selected for this pilot because it was the
end of the year and the examination was intended for third graders at the
beginning of the year.

The
materials used for the examination were the following: student answer sheets, a
portable overhead projector, 4 large white poster boards for a screen and a
chronometer. The answer sheets were specially designed so that there was no
room for written computation and to discourage students from copying. The
examination was given by the investigator. An external evaluator did the timing
for the test and did the test correction.

Before the second pilot testing, the instrument was
given to a retired elementary school principal for Spanish grammatical revision
and test evaluation commentary. The main change recommended was to not present
the missing addends problems with a blank but instead with a picture in the
place of the blank, because she considered that it would be confusing for the
students. She also suggested that numbers be placed next to the sets of objects
presented on the transparency, so that the students would not necessarily have
to count them.

The second pilot examination was given to 140 third
grade students at a local public school in the Ponce, Puerto Rico. It was
administered at the beginning of the school year.

An item analysis test (Iteman) was applied to
measure difficulty and discrimination of examination items. According to these
indexes particular items were eliminated, modified or left on the test (see
Appendix B and C for Iteman test interpretation scale and table sample).

During each stage of the examination revision an
external evaluator who is a mathematics professor who is a working statistician
in a university was consulted. The final version was used as the pre and post
treatment measure (Appendix D).

After
administration the pre and posttest
results were analyzedto
obtain the following data:

An analysis for each
student of gain or loss with respect to the pretest and the posttest.

A t-test for independent
samples to see if there was a significant difference among the means of
the pretest and posttest for each group. (low achievers and high
achievers).

An analysis of variance
(ANOVA) to determine statistically significant difference between group
types.

The interview questions for the teachers were
selected as conversation openers so that the teachers would feel at ease with
the investigator but at the same time begin the process of discovering their
beliefs and understanding of mental computation, number sense, and other areas
of pedagogy (Appendix E). The first questionnaire was constructed using the
same questions that were included in the interview as confirmation (Appendix
F). The second questionnaire included questions about the teachers background,
in-service training, and questions about pedagogical strategies that they were
using during this year (Appendix G). They were also asked about their
participation in the study. The third questionnaire was constructed using
statements about assessment that included some common myths or beliefs about
assessment as pointed out by NCTM (1991) (Appendix H). Teachers were asked to
write what they thought of each statement. The questions for the last interview
were similar to those of the first interview but also included selected
questions from the three questionnaire for purpose of comparison (Appendix I).
They also were asked to evaluate the process of their participation in the
study. A summary of the teachers' shared beliefs obtained through the analysis
of the interviews and the questionnaires was prepared.

The questions for the clinical interview for the
students were constructed in the following manner:

1.Eight
addition, five subtraction and one mixed operations problem types were selected
from the criterion referenced table used to construct the pre/posttest
(Appendix J),

Problems parallel to
those on the examination for the corresponding criteria were constructed,

A conservation problem was designed to checkif the students were in the
concrete operational stage according to Piaget's theory of cognitive
development. Each student was shown eight colored markers and eight rubber
bands placed in two parallel rows so that each marker had a rubber band
next to it. After the student had enough time to observe both sets, the
chips and the rubber bands were removed and placed again on the table.
This time the rubber bands were placed closer together and the markers were
placed the same way as before. The student was asked if there were more
rubber bands or markers, or was there the same amount. The purpose of the
exercise is to determine if the student understands the law of
conservation which is indicative that the child is in the concrete
operational stage (which is usual for the third grade students).

Recording materials
were prepared and a table was designed to analyze the types of strategies
the students used (Appendix K),

Visual aids of each
problem were prepared (see Appendix L).

To analyze the results, of the clinical interviews
of the students, the percent of use of inefficient, standard and nonstandard
procedures were compared. This clasification is defined in the following
manner:

Standard: The student
uses a mental version of a pencil and paper algorithm or uses recall of
basic facts.

Nonstandard with no
reformulation: The left to right process is used. (38+45=30+40+13)

Nonstandard with
reformulation: the numbers are reformulated to make the computation easier
(38+45= 40+45-2).

Inefficient invented
strategies: the students use either a left to right or a right to left
process but combine the place values incorrectly. (23 + 36 = 3+3+2+6).

Percents of students using standard and non-standard
computational strategies were also computed. A clasification of the types of
nonstandard strategies with and without reformulation was computed.

Theoretical
Basis and Description

Throughout much of the literature reviewed in Chapter
2 on mathematics learning, a common theme kept revealing itself. This theme was
the need for children to "invent" their own ways of "doing"
mathematics. The activities for the treatment in this study were developed with
a constructivist framework. Mental computation strategies were not taught.
Throughout the activities, students were motivated to share their strategies,
evaluate their own work, visualize and write about their thinking, in order to promote the development of
number sense and mental computation skills.

The identification of patterns was a theme that was
included in most of the activities for the treatment. There was also emphasis
on connecting ideas presented in the different activities. Various visual
models were used to help students develop number and number properties
visualization skills. The models used were dominoes, playing cards, ten frames
and painted wooden sticks (a variation of bean sticks). Game-like activities
were frequently used. Place value concepts and addition of two digit numbers
were developed using base-ten blocks. The hundreds table was used for pattern
identification, addition and subtraction.

In order to begin the development of the materials for
this study a database of games and instructional activities from diverse
resources was collected. The database
and thestudies on addition,
subtraction and counting were the starting points for the materials
development.

The central themes to be included were patterns,
computation using concrete and pictorial models, place value and computation
using the 0 to 99 table (see Appendix M for Curriculum Plan). Using a list of
mental computation skills developed through the review of the literature (see
Appendix N) and the central themes, the activities were developed.

For the evaluation of the activities, the
investigator developed an instrument to evaluate the material through classroom
observation or by reading the activity (Appendix O). The instrument was
developed originally in the form of a checklist based on one by Caine and Caine
(1994) for use in the evaluation in the activities for the treatment. It was
redesigned and given to a panel of four professors and a retired principal with
an open questionnaire for evaluation purposes. After the copies of the
instrument were returned, the suggestions were incorporated and the instrument
was given a trail run in the evaluation the first five activities. This was
done by a retired elementary school principal. These were evaluated and the
instrument was also evaluated for the purpose it was used. Again the
suggestions were incorporated and the instrument was used by two external
evaluators, to evaluate the activities developed for the treatment in the
study. Their suggestions were incorporated into the final design of the
activities (Appendix P). The instrument was also used by the investigator and
the treatment teacher for observation and evaluation of activities in class.
These evaluations were useful in the refining of the activities in the
classroom setting.